The purpose of this project is to utilize an Hypothesis-testing procedure that permits a trial-by-trial assessment of why a response is made in order to study the development of concepts related to an understanding of proportion. We shall determine which cognitive strategies are available to children at specific stages of development, how these strategies are put to use in solving proportion and probability tasks, what the child does when a strategy does not work, and what limits are imposed by the nature of the child's cognitive system on his ability to generate and test alternate strategies. The cognitive processes related to proportion will be studied in children from age four to adulthood. Experiments will examine the effects of varying the number and arrangement of elements in a set as well as the effects of theory-based training techniques on children's performances on the proportionality tasks. The relationship between cognitive level of functioning and the strategies used by children of different ages will also be determined. Another purpose of this research is to study the generality of proportional reasoning. Children's performances on this task will be related to a variety of tasks, including probability learning, chance, and other tasks involving proportional reasoning, enabling us to investigate the generality of the presumed underlying cognitive structure proposed by Inhelder and Piaget, the "proportionality scheme." Also, the development of hypothesis testing in children will be studied in an area other than discrimination learning so that the generality of previous findings in this area can be ascertained.